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We present an efficient convolution kernel for Convolutional Neural Networks (CNNs) on unstructured grids using parameterized differential operators while fo- cusing on spherical signals such as panorama images or planetary signals. To this end, we replace conventional convolution kernels with linear combinations of differential operators that are weighted by learnable parameters. Differential oper- ators can be efficiently estimated on unstructured grids using one-ring neighbors, and learnable parameters can be optimized through standard back-propagation. As a result, we obtain extremely efficient neural networks that match or outper- form state-of-the-art network architectures in terms of performance but with a significantly smaller number of network parameters. We evaluate our algorithm in an extensive series of experiments on a variety of computer vision and climate science tasks, including shape classification, climate pattern segmentation, and omnidirectional image semantic segmentation. Overall, we (1) present a novel CNN approach on unstructured grids using parameterized differential operators for spherical signals, and (2) show that our unique kernel parameterization allows our model to achieve the same or higher accuracy with significantly fewer network parameters.